ECLAIR 1.18

How ECLAIR graphs and trees are generated

ECLAIR stands for Ensemble Clustering for Lineage Analysis, Inference and Robustness. It proceeds as follow:

• Choose among affinity propagation, hierarchical or k-means clustering and DBSCAN (cf. our DBSCAN_multiplex and Concurrent_AP packages for streamlined and scalable implementations of DBSCAN and affinity propagation clustering) for how to group cells from subsamples of your dataset.

• Such a subsample is obtained by density-based downsampling (as implemented in our Density_Sampling software posted on the Python Package Index), either by aiming for an overall number of datapoints to extract from the dataset or by specifiying a target percentile of the distribution built from local densities around each datapoint.

• ECLAIR then goes about performing several rounds of downsampling and clustering on such subsamples, for as many iterations as specified by the user. After each run of clustering a given subsample, the datapoints that were left over by the downsampling procedure are upsampled by associating them to the closest centroid in high-dimensional feature space.

• For each such run, build a minimum spanning tree. This minimum spanning tree is obtained from a matrix of L2 pairwise similarities between the centroids associated to each cluster.

• The next step obtains a consensus clustering from this ensemble of partitions of the whole dataset. Three heuristic methods are considered for this purpose: CSPA, HGPA and MCLA, all of them based on graph or hypergraph partitioning (cf. the documentation of our Cluster_Ensembles package for more information).

• Once a consensus clustering has been reached, we build a graph from the consensus clusters and from the information associated with the ensemble of partitions from which those consensus clusters have just been computed. The edge weights of this graph are calculated as the mean of the following distribution: for each of the 2-uple consisting of one datapoint from consensus cluster a and another datapoint from consensus cluster b, scan over the ensemble of partitions and keep track of the distance separating those two samples across each partition comprising the cluster ensemble.

• We then obtain a minimum spanning tree from this graph, for convenience of visualization as well as for later comparison with a few other methods that purport to provide estimates of cell lineages (including the popular SPADE method, whose reproducibility issues spurred the development of ECLAIR. A module from the present package is indeed dedicated to illustrating the superior statistical performance of ECLAIR).

Statistical performance of ECLAIR

To compare two lineage trees, one has to take into account their edge connections but also the sample contents of their nodes, since the variation associated to subsampling results in different clusters of samples. Although there are many papers on graph matching and graph comparison, we are not aware of any previously published method that takes into account the node differences. We therefore developed customized statistical tests suitable for comparing lineage trees.

• The first score we developped aims to compare the overall similarity between two lineage trees, T_1 and T_2. For each tree, we evaluate the path length between every pair of cells in the population, based on the edge connectivity. The correlation between the two sets of path length values is used as a score to compare the overall similarity of T_1 and T_2. For a moderately large dataset of 500,000 samples, this would naively translate into more than 100 billion pairs of distances along T_1and along T_2. The details of the much more efficient algorithm we developped for that purpose is available from the docstrings of our package; the gist of this algorithm is to first build a contingency table recording the overlap in the number of samples between pairs of T_1 nodes versus pairs of T_2 nodes.

• Second, we define D_ijas an edge-specific measures of statistical dispersion to evaluate the robustness of each edge within a given lineage tree , denoted T*. Specifically, for each edge E_ij connecting a pair of clusters C_i* and C_j*, we define the dispersion D_ij associated with E_ij as the standard deviation of the the distribution of path lengths L^a(x,y), where x and y are selected from C_i*and C_j* respectively, and a is summed over the partitions and minimum spanning trees from the ensemble out of which T* was constructed in the first place. This distribution is the same as the distribution of path lengths whose mean was used to assign a weight to edge E_ij of the graph from which the ECLAIR tree was inferred in the first place.

• The afore-mentioned measure of statistical dispersion is computed solely in terms of the partitions and trees making up an ensemble from which a consensus clustering and an ECLAIR tree are then extracted. We also compare this measure with another measure of statistical dispersion, obtained by independently generating 50 different ECLAIR trees in a procedure reminiscent of the bootstrap. One such tree is singled out as a reference tree. For each edge of this reference tree, we keep track across the rest of the 49 ECLAIR tree how spread out are the pairs of cells comprising the two nodes of this reference edge.

Our ECLAIR package features a module entirely devoted to computing through suitable data structures and algorithms such statistical measures and a few more tests on pairs of ECLAIR trees.

Installation

ECLAIR is written in Python 2.7. It has been tested on Fedora Linux and on Ubuntu and should be supported by any other member of the UNIX-like family of operating systems.

Install ECLAIR by sending a request to the Python Package Index (PyPI) as follows:

• start a terminal;

• enter pip install ECLAIR.

Any missing or out-of-date dependency should be automatically resolved. Apart from the Python Standard Library, those include:

• Cluster_Ensembles (version 1.16 or later)

• Concurrent_AP (version 1.4 or later)

• DBSCAN_multiplex (version 1.5 or ulterior)

• Density_Sampling (1.1 or subsequent version)

• igraph

• matplotlib (version 1.4.3 at least)

• munkres

• numpy (1.9.0 or ulterior version)

• scipy (0.16 or later version)

• sklearn

• setuptools

• tables

Please note that as part of the installation of this package, some code written in C that is part of the Cluster_Ensembles package will be automatically compiled, under the hood and according to the specifications of your machine. For this process to go seamlessly, you have however to ensure availability of CMake and GNU make on your operating system. Cluster_Ensembles also requires the 32-bit version of the GNU C library. Please refer to the Cluster_Ensembles documentation for more information on how to meet those few requirements depending on Linux distribution.

Usage

To subject a dataset to an ECLAIR analysis:

• start a terminal;

• enter python -m ECLAIR.Build_instance [options] [file_name], where file_name denotes the path to the data about to be processed.

It is generally recommended to leave the options and file_name field empty, which will trigger an interface asking the user to provide the path to the dataset to be processed and some guidance on the choice of parameters for the ECLAIR analysis at hand. Each row of the dataset accessed via the path file_name must correspond to a sample, whose features must be on display in a tab-separated format. A folder will be created in your current working directory, containing information on your ECLAIR tree and the underlying weighted graph (such as its adjacency matrix and confidence coefficients for each edge) along with a PDF figure illustrating a force-directed representation of the inferred lineage tree.

To launch a full-fledged statistical performance analysis of ECLAIR and see how it consistenly performs better than SPADE, a popular method for estimating cell lineages, proceed as follows:

• at the Shell command-line interface or graphical user interface, type in python -m ECLAIR.Statistical_performance.

The eponymous folder ECLAIR_performance will be created in your current directory, recording on the fly the results of various statistical tests and comparisons of ECLAIR graphs and trees, as well as of SPADE trees.

In the current version, the statistical performance of ECLAIR is only evaluated for a fairly large (by the current standards of computational biology) flow cytometry dataset of half-a-million samples and 8 features, as well as on a qPCR dataset of mouse bone marrow samples. It shouldn’t be difficult for anyone competent in Python to quickly peruse through the source code of ECLAIR and bring about a few of the changes required to submit his/her own data to a similar statistical analysis (those changes mostly pertain to domain-specific knowledge and to the format of your dataset). ECLAIR has been designed so as to accommodate arbitrarily large datasets (this is achieved through the use HDF5 data structures, most notably).

Upon sending the ECLAIR_performance command, several “experiments” will be performed, including the comparisons of pairs of ECLAIR graphs or trees and pairs of SPADE trees generated on the same dataset. The comparison of ECLAIR instances and of SPADE instances generated on non-overlapping datasets and evaluated on a separate test set calls for detailed explanations.

We are splitting a dataset into three equally-sized, non-overlapping parts, S1, S2 and S3. We train an ECLAIR tree (Ecl_1) and a SPADE tree on S1 (Spd_1). We then train another ECLAIR tree (Ecl_2) and yet another SPADE tree (Spd_2) on the set S2.

The training procedure for Ecl_1 involves 50 runs of downsampling and clustering of the samples within S1. The downsampling ratio is set at 50%. Therefore, Ecl_1 is an aggregation of 50 trees, all generated from S1 alone.

In order to compare Ecl_1 with Ecl_2, the cells in S3 are mapped to the clusters/nodes in Ecl_1 and in Ecl_2 to which they are nearest in the high-dimensional gene expression space.

Idem when it comes to comparing Spd_1 and Spd_2.

The procedure outlined above is repeated 10 times. We end up with two lists of 30 correlation coefficients telling us about the similarity of as many pairs of ECLAIR or SPADE trees. Indeed, while things have been exposed as involving only the evaluation of Ecl_1 and Ecl_2 on S3 using as a test set, one can also generate an ECLAIR tree using S3 as a training set. This allows the additional comparisons of Ecl_1 with Ecl_3 and of Ecl_2 with Ecl_3.

It also bears pointing out we are using the same test set (S3) for assessing the similarity of pairs of ECLAIR trees (Ecl_1 vs. Ecl_2) as for evaluating the similitude of pairs of SPADE trees (Spd_1 vs. Spd_2).

References

• Giecold, G., Marco, E., Trippa, L. and Yuan, G.-C., “Robust Inference of Cell Lineages”. ArXiv preprint [q-bio.QM, stat.AP, stat.CO, stat.ML]: http://arxiv.org/abs/1601.02748

• Strehl, A. and Ghosh, J., “Cluster Ensembles - A Knowledge Reuse Framework for Combining Multiple Partitions”. In: Journal of Machine Learning Research, 3, pp. 583-617. 2002

• Conte, D., Foggia, P., Sansone, C. and Vento, M., “Thirty Years of Graph Matching in Pattern Recognition”. In: International Journal of Pattern Recognition and Artificial Intelligence, 18, 3, pp. 265-298. 2004